The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

The partition corresponding to the even cycles.
A cycle of length $\ell$ of a signed permutation $\pi$ can be written in two line notation as
$$\begin{array}{cccc} a_1 & a_2 & \dots & a_\ell \\ \pi(a_1) & \pi(a_2) & \dots & \pi(a_\ell) \end{array}$$
where $a_i > 0$ for all $i$, $a_{i+1} = |\pi(a_i)|$ for $i < \ell$ and $a_1 = |\pi(a_\ell)|$.
The cycle is even, if the number of negative elements in the second row is even.
This map records the integer partition given by the lengths of the odd cycles.
The integer partition of even cycles together with the integer partition of the odd cycles determines the conjugacy class of the signed permutation.

Return the signed permutation with all signs reversed.

The signed permutation with all signs positive.